Results 1  10
of
30
Randomness, relativization, and Turing degrees
 J. Symbolic Logic
, 2005
"... We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompre ..."
Abstract

Cited by 38 (16 self)
 Add to MetaCart
We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompressible: C(x) c. The `only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of timebounded Ccomplexity.
Computing a glimpse of randomness
 Experimental Mathematics
"... A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In ..."
Abstract

Cited by 20 (10 self)
 Add to MetaCart
A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 63 bits of a Chaitin Omega: 000000100000010000100000100001110111001100100111100010010011100. Full description of programs and proofs will be given elsewhere. 1
Chaitin Ω Numbers, Solovay Machines, and Incompleteness
 COMPUT. SCI
, 1999
"... Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay's construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show ..."
Abstract

Cited by 17 (15 self)
 Add to MetaCart
Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay's construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show that every c.e. random real is the halting probability of a universal Chaitin machine for which ZFC cannot determine more than its initial block of 1 bitsas soon as you get a 0 it's all over. Finally, a constructive version of Chaitin informationtheoretic incompleteness theorem is proven.
Optimization Is Easy and Learning Is Hard In the Typical Function
, 2000
"... Elementary results in algorithmic information theory are invoked to show that almost all finite functions are highly random. That is, the shortest program generating a given function description is rarely much shorter than the description. It is also shown that the length of a program for learning o ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
Elementary results in algorithmic information theory are invoked to show that almost all finite functions are highly random. That is, the shortest program generating a given function description is rarely much shorter than the description. It is also shown that the length of a program for learning or optimization poses a bound on the algorithmic information it supplies about any description. For highly random descriptions, success in guessing values is essentially accidental, but learning accuracy can be high in some cases if the program is long. Optimizers, on the other hand, are graded according to the goodness of values in partial functions they sample. In a highly random function, good values are as common and evenly dispersed as bad values, and random sampling of points is very efficient.
From Heisenberg to Gödel via Chaitin
 INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS
, 2004
"... In 1927 Heisenberg discovered that the "more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa". Four years later ..."
Abstract

Cited by 11 (9 self)
 Add to MetaCart
In 1927 Heisenberg discovered that the "more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa". Four years later
A Characterization of C.E. Random Reals
 THEORETICAL COMPUTER SCIENCE
, 1999
"... A real # is computably enumerable if it is the limit of a computable, increasing, converging sequence of rationals; # is random if its binary expansion is a random sequence. Our aim is to offer a selfcontained proof, based on the papers [7, 14, 4, 13], of the following theorem: areal is c.e. and ra ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
A real # is computably enumerable if it is the limit of a computable, increasing, converging sequence of rationals; # is random if its binary expansion is a random sequence. Our aim is to offer a selfcontained proof, based on the papers [7, 14, 4, 13], of the following theorem: areal is c.e. and random if and only if it a Chaitin# real, i.e., the halting probability of some universal selfdelimiting Turing machine.
Practical implications of new results in conservation of optimizer performance
 Parallel Problem Solving from Nature  PPSN VI, 6th International Conference
, 2000
"... Abstract. Three theoretical perspectives upon conservation of performance in function optimization are outlined. In terms of statistical information, performance is conserved when the distribution of functions is such that all domain subsets of a given size have identically distributed random values ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Abstract. Three theoretical perspectives upon conservation of performance in function optimization are outlined. In terms of statistical information, performance is conserved when the distribution of functions is such that all domain subsets of a given size have identically distributed random values. In terms of algorithmic information, performance is conserved because almost all functions are algorithmically random. Also, an optimizer’s algorithmic complexity bounds the information gained or lost in its processing of the test function. The practical consequences of these theoretical results are explored, with emphasis upon the results from algorithmic information theory, which are new. 1
Mathematical proofs at a crossroad
 Theory Is Forever, Lectures Notes in Comput. Sci. 3113
, 2004
"... Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimen ..."
Abstract

Cited by 7 (7 self)
 Add to MetaCart
Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomaticdeductive proofs are not a posteriori work, a luxury we can marginalize nor are computerassisted proofs bad mathematics. There is hope for integration! 1
Gordon Pask's Conversation Theory: A Domain Independent . . .
"... Althoct it isco)"R() (as argued by many) that distinct knoinct1 doinc do present particularproicul o coic to knoi in this paper it is argued that it ispoPP(;' (and useful)to coul)1), ado;(; independentmode o theproq"",; o coq" to knoq oo in whichoich1)q) share understandings anddo so in agreed ways. ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Althoct it isco)"R() (as argued by many) that distinct knoinct1 doinc do present particularproicul o coic to knoi in this paper it is argued that it ispoPP(;' (and useful)to coul)1), ado;(; independentmode o theproq"",; o coq" to knoq oo in whichoich1)q) share understandings anddo so in agreed ways. Themo1P in questio is parto theco);q;;1.q) theo; (CT)o Go)1q Pask. CT, as atheo, o theo co,1.q)(q andco,(qq)1.,;qq has particular relevancefo fovance1P(( issues in science and scienceeducatioR CT explicitlyprolicit a "radicalcoical1P))R1. (RC) epistemo)1., A briefacco1) is giveno the main tenetso RC and CT's place in that traditio and the traditio. o cybernetics. The paper presents a briefnof1P),R1., acco1o o the maincon1)P o CT includingelabong1q(; by Laurillard and HarriAugstein andTho)"q As parto CT, Paskalso elabo,",1 ametho1.,( kno;P;1. and task analysisfo analysing the structureo different knoerent doeren this metho','1 is sketched ino1,,)RR Keywo)RR Co)RR1.)'P theo)R epistemo...